quasicompact morphism

The modifier **‘quasi-compact’** (or simply **‘quasicompact’**) is used to denote a compactness property in the relative setup (that is, for morphisms) and in the setups emphasising non-Hausdorff topology. For example, schemes over complex numbers have both the complex analytic topology and the Zariski topology; we use “quasicompact” for Zariski and “compact” for complex topology in that setup.

For many topologists, analysts and so on, the word ‘compact’ means a compact Hausdorff space. Some topological schools distinguish a “compact space” (which is not necessarily Hausdorff) and a “compactum” (which is Hausdorff); and similarly a “paracompact space” and a “paracompactum”. In algebraic geometry, in contrast, one usually says **quasicompact space** to denote a topological space which is compact but not necessarily Hausdorff. For example, the Zariski topology on an algebraic variety and the topology of an étalé space of a sheaf (even over a Hausdorff topological space) are typically not Hausdorff.

A scheme is quasicompact iff it has a Zariski cover by finitely many open affine subschemes. In particular, any affine scheme is quasicompact.

Most important is the relative version of this concept. A morphism $f\colon X \to Y$ of schemes is a **quasicompact morphism** if the inverse image of a quasicompact Zariski open subset of $Y$ is quasicompact (EGAI6.6.1).

It is straightforward to show [EGAI6.6.4] that it is enough to require this for affine subsets of $Y$, or even to require the existence of a single covering $Y = \cup_i U_i$ of $Y$ by open affine subschemes $U_i$, such that the inverse image $U_i \times_Y X$ of $U_i$ in $X$ is quasicompact. A scheme $X$ over a base scheme $S$ is quasicompact if the canonical morphism $X \to S$ is quasicompact. This is consistent, because if $X$ is a usual scheme (over the spectrum of integers $S = \mathbb{Z}$) or, more generally, a relative scheme over an *affine* scheme $S$, quasicompactness of the canonical morphism $X\to S$ is by the above criteria clearly equivalent to the usual quasicompactness of $X$.

A composition of quasicompact morphisms is quasicompact, and the pullback of quasicompact morphisms is quasicompact. This enables the definition for algebraic stacks: a morphism of algebraic stacks is quasicompact if the pullback of that morphism to some atlas is quasicompact.

Algebraic geometers sometimes (but more rarely) also talk about **quasicompact objects** in more general categories, meaning compact objects (object which corepresent covariant functors commuting with filtered colimits); with or without a modifier denoting a cardinal ($\kappa$-quasicompact objects).

An old discussion on the terminological aspects (Mike, Zoran, Toby) is at $n$Forum here.

category: algebraic geometry

Revised on February 3, 2014 15:19:07
by Urs Schreiber
(89.204.130.154)